The Canonical Generalized Polar Decomposition
Nicholas John Higham, Christian Mehl, Françoise Tisseur · SIAM Journal on Matrix Analysis and Applications · 2010
The polar decomposition of a square matrix has been generalized by several authors to scalar products on $\mathbb{R}^n$ or $\mathbb{C}^n$ given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition $A = WS$, defined for general $m\times n$ matrices A, where W is a partial $(M,N)$-isometry and S is N-selfadjoint with nonzero eigenvalues lying in the open right half-plane, and the nonsingular matrices M and N define scalar products on $\mathbb{C}^m$ and $\mathbb{C}^n$, respectively. We derive conditions under which a unique decomposition exists and show how to compute the decomposition by matrix iterations. Our treatment derives and exploits key properties of partial $(M,N)$-isometries and orthosymmetric pairs of scalar products, and also employs an appropriate generalized Moore–Penrose pseudoinverse. We relate commutativity of the factors in the canonical generalized polar decomposition to an appropriate definition of normality. We also consider a related generalized polar decomposition $A = WS$, defined only for square matrices A and in which W is an automorphism; we analyze its existence and the uniqueness of the selfadjoint factor when A is singular.